Apparatus and method for detecting optical profile

ABSTRACT

Apparatus and methods for detecting optical profile are disclosed herein. In one embodiment, an apparatus includes a laser, a beam splitter, a collimation optical unit, first and second holders respectively holding a first test flat mirror and a second test flat mirror, a phase shifter connected with the first holder, and an angular measurement unit for measuring an angular error of the first test flat mirror and the second test flat mirror on the two holders. The first test flat mirror has a first test flat and the second test flat mirror has a second test flat. The apparatus further includes a planar imaging unit for generating the interfered test light having a direction generally along an x-axis direction of the first test flat and an x-axis direction of the second test flat and a convergence optical unit for projecting the interfered test light onto a detector.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims priority to Chinese Application No.201010266732.7, filed on Aug. 24, 2010, the disclosure of which isincorporated herein by reference in its entirety.

TECHNICAL FIELD

The present disclosure relates generally to apparatus and methods foroptical profile detection.

BACKGROUND

High accuracy interferometers have been widely used for surfacemeasurement not only in optical manufacturing but also in new fieldssuch as optical disk surface measurement and semiconductor crystal planemeasurement. More and more emphasis has been placed on detectionaccuracy of peak-to-valley (“PV”) values in the sub-nanometer range.FIG. 8 schematically shows a system configured according to aconventional six-step absolute flatness detection technique. As shown inFIG. 8, the system comprises a laser light source 1 a, a collimationoptical system 3 a and 4 a, a beam splitter 2 a, a front surface 5 a ofa first test flat, a piezoelectric transducer (“PZT”) phase shifter 6, afront surface 7 a of a second test flat, a convergence optical system 8a, a charge-coupled device (“CCD”) 9 a, and a computer 10 a.

FIG. 9 schematically shows a measurement process of the system in FIG.8. The measuring process includes:

-   -   (1) measuring an optical path difference between the front        surface of the first test flat and the front surface of the        second test flat;    -   (2) rotating the first test flat by 180° from its original        position, and measuring an optical path difference between the        front surface of the first test flat and the front surface of        the second test flat;    -   (3) rotating the first test flat by 90° from its original        position, and measuring an optical path difference between the        front surface of the first test flat and the front surface of        the second test flat;    -   (4) rotating the first test flat by 45° from its original        position, and measuring an optical path difference between the        front surface of the first test flat and the front surface of        the second test flat;    -   (5) replacing the second test flat with a third test flat, and        measuring an optical path difference between the front surface        of the first test flat and the front surface of the third test        flat;    -   (6) replacing the first test flat with the second test flat, and        measuring an optical path difference between the front surface        of the second test flat and the front surface of the third test        flat.        The first test flat, the second test flat, and the third test        flat are calculated based on the measurement results. The        foregoing six-dimensional angular rotation platform typically        has an accuracy of about 10 micro-radians to about 1        milli-radian. Such accuracy levels may not satisfy the ever        increasing accuracy requirements in today's industry.        Accordingly, there is a need for apparatus and methods that can        improve the measurement accuracies of conventional surface        measurement platforms.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of an apparatus for optical profiledetection according to embodiments of the present technology.

FIG. 2 is a schematic diagram of an apparatus not containing a planarimaging unit according to embodiments of the present technology.

FIG. 3 schematically shows a process for detecting an optical profileaccording to embodiments of the present technology.

FIG. 4 is a flowchart illustrating a method for detecting an opticalprofile according to embodiments of the present technology.

FIG. 5 is a schematic diagram showing test flats of test flat mirrors.

FIG. 6 schematically shows interference between a first test flat and asecond test flat.

FIG. 7 schematically shows interference between a first test flat and asecond test flat when having a consistent x-axis direction with an addedplanar imaging unit.

FIG. 8 schematically shows an apparatus for the conventional six-stepabsolute measurement technique.

FIG. 9 schematically shows a measurement process of the conventionalsix-step absolute measurement technique.

FIGS. 10 a, 10 b, and 10 c are diagrams showing simulation images ofZernike fit according to embodiments of the present technology.

FIGS. 11 a, 11 b, and 11 c are diagrams showing differences between thesimulation result and a true profile by assuming an angular rotationerror according to embodiments of the present technology.

FIGS. 12 a, 12 b, and 12 c are diagrams showing differences between thesimulation result and a true profile by assuming the same angularrotation error according to the conventional six-step absolutemeasurement technique.

DETAILED DESCRIPTION

Various embodiments of systems, apparatus, and associated methods foroptical profile detection are described below. A person skilled in therelevant art will also understand that the technology may haveadditional embodiments, and that the technology may be practiced withoutseveral of the details of the embodiments described below with referenceto FIGS. 1-7 and 10 a-12 c.

FIG. 1 is a schematic diagram of an apparatus for optical profiledetection according to embodiments of the present technology. As shownin FIG. 1, the apparatus includes a laser 1, a beam splitter 2, acollimation optical unit 3 and 4, a first holder 5, a phase shifter 6, aplanar imaging unit 7, a second holder 8, an angular measurement unit 9,a convergence optical unit 10, a CCD detector 11, and a computer 12operatively coupled to one another.

The laser 1 is configured as a light source to emit a laser beam. Thecollimation optical unit 3 and 4 is configured to collimate the laserbeam emitted by the laser 1, and thus forming a generally uniformilluminated area. The beam splitter (or splitter) 2 is configured totransmit the laser beam emitted by the laser 1 as an illuminating lightand a reflecting interfered test light to a test flat on the holder 5.The phase shifter 6 is controlled by the computer 12 and is configuredto shift a phase of lights passing through the test flat. Theconvergence optical unit 10 is configured to project the interfered testlight onto the CCD detector 11.

The angular measurement unit 9 is configured to measure an angular errorof a lens on the first holder 5 and a lens on the second holder 8. Theplanar imaging unit 7 is configured to generate the interfered testlight along the x-axis of the first test flat and the x-axis of thesecond test flat. FIG. 7 schematically shows the interference betweenthe first test flat and the second test flat when their x-axes arealigned with an added planar imaging unit.

FIG. 2 shows the apparatus in FIG. 1 with the planar imaging unit 7removed during a measuring process according to embodiments of thepresent technology. In one embodiment, the computer 12 can controllablyremove the planar imaging unit 7 from between the two holders 5 and 8.The laser 1 emits a laser beam that passes through the beam splitter 2and the collimation optical unit 3 and 4 to produce the illuminatinglight.

The computer 12 also controls the phase shifter 6 to shift phase. Theangular measurement unit 9 is configured to measure an angular error ofthe test flat on the first holder 5 and the test flat on the secondholder 8. The laser 1 is placed on a front focus of the beam splitter 2and the collimation optical unit 3 and 4. The light passing through thecollimation optical unit 3 and 4 is approximately parallel. The testflat on the first holder 5 is placed behind the collimation optical unit3 and 4, and has its center of optical axis generally aligned with thecenter of the collimation optical unit and the center of the beamsplitter 2. The test flat on the second holder 8 is placed behind andparallel to the lens on the first holder 5. The phase shifter 6 isconnected with the first holder 5 for controlling phase shift of thetest flat on the first holder 5.

Interference occurs between the test flat on the first holder 5 and thetest flat on the second holder 8. The interfered light returns along anoptical path passing through the beam splitter 2 and converges by theconvergence optical unit 10 to the CCD detector 11. The CCD detector 11receives the interfered light and provides information of the interferedlight to the computer 12, which in turn calculates an optical pathdifference based on the received information.

The computer 12 can also controllably insert the planar imaging unit 7between the two holders 5 and 8. The face of the planar imaging unit 7can be generally parallel to the first test flat and the second testflat. The phase shifter 6 is connected with the first holder 5 and isadapted for controlling the phase shift of the first test flat on thefirst holder 5.

FIG. 3 schematically shows relative positions of the two test flats inthe apparatus in FIGS. 1 and 2 during a measurement process fordetecting an optical profile according to embodiments of the presenttechnology. FIG. 4 is a flowchart showing the measurement process, andFIG. 5 is a schematic diagram showing the test flat mirrors during themeasurement process. As shown in FIG. 4, in certain embodiments, themeasurement process adopts an absolute measurement technique, which cansimultaneously measure a profile error of the test flats of the Fizeauinterferometer by cross detection of three flats. In other embodiments,other suitable measurement techniques can also be adopted.

In the embodiment illustrated in FIG. 4, for flat mirrors, as shown inFIG. 5, the flats to be measured are defined as test flats. The testflats of three test flat mirrors are respectively numbered as a firsttest flat, a second test flat, and a third test flat. The first testflat mirror is placed in the second holder 8 and the second test flatmirror is placed in the first holder 5, so that the second test flatfaces the first test flat. Position information of the first test flatand the second test flat in an x-axis direction and a y-axis directionis recorded.

A current position of the first test flat is defined as its originalposition. An optical path difference between the first test flat and thesecond test flat is measured. light emitted by the laser 1 passesthrough the beam splitter 2 and then through the collimation opticalunit 3 and 4 and illuminates the second test flat on the first holder 5,and is reflected to form a reference light. Light passing through thesecond test flat on the first holder 5 and illuminating the first testflat on the second holder 8 is reflected and interfered with thereference light to form test light.

The test light produced by interference between the light reflected bythe second test flat on the first holder 5 and the light reflected bythe first test flat on the second holder 8 converges on the CCD detector11 via the collimation optical unit 3 and 4 and the convergence opticalunit 10, to form an interference pattern, which is recorded by the CCDdetector 11 and then is stored and processed by the computer 12. Thephase shifter 6 is connected to the first holder 5, and is adapted forcontrolling the phase shift of the second test flat on the first holder5 to produce a plurality of interference patterns.

The optical path difference information can be obtained by processingthe interference patterns as follows:

M ₁ =A+B ^(x),

where A represents profile information of the first test flat, Brepresents profile information of the second test flat; A and B arefunctions of x, y coordinates, i.e., A=A(x,y), B=B(x,y). M₁ representsan optical path difference between the first test flat and the secondtest flat in the first interference measurement.

FIG. 6 schematically shows relative positions of the first test flat andthe second test flat when the interference measurement is performed. Anx-axis direction of the first test flat is defined as a positivedirection, and when the first test flat opposes the second test flat,the x-axis of the second test flat is reversed about a y-axis, and thusB^(x)=B(−x, y).

Subsequently, during stage S2, the first test flat is rotated clockwisefrom its original position in stage S1 by 180° and the position of thesecond test flat is unchanged. An optical path difference between thefirst test flat and the second test flat is measured. Because the secondholder has a rotation error, an angle value of a current position of thefirst test flat with respect to its original position during stage S1 ismeasured by an angular measurement unit 9, which is then subtracted by180° to obtain an angular rotation error Δθ1 for stage S2. Thus, M₂ iscalculated as follows:

M ₂ =A ^(180°+Δθ1) +B ^(x),

where A^(180°+Δθ1) represents profile information of the first test flatafter being rotated clockwise by 180°+Δθ1 degrees from its originalposition in stage S1. B^(x) represents the profile information of thesecond test flat after its x-axis is reversed about the y-axis. M₂represents, after interference occurs between the first test flat andthe second test flat, an optical path difference between the first testflat, after being rotated clockwise by 180°+Δθ1 degrees from itsoriginal position in stage S1, as shown in FIG. 3.

During stage S3, the first test flat is rotated counterclockwise by 90°from its position in stage S2 and the position of the second test flatis unchanged. An optical path difference between the first test flat andthe second test flat is measured. Because the second holder has arotation error, an angle value of a current position of the first testflat with respect to its original position in stage S1 is measured,which is then subtracted by 90° to obtain an angular rotation error Δθ2of stage S3. M₃ can be calculated as follows:

M ₃ =A ^(90°+Δθ2) +B ^(x)

where A^(90°+Δθ2) represents profile information of the first test flatafter being rotated clockwise from its original position in stage S1 by90°+Δθ2 degrees. B^(x) represents the profile information of the secondtest flat after its x-axis being reversed about the y-axis. M₃represents, after interference occurs between the first test flat andthe second test flat, an optical path difference between the first testflat, after being rotated clockwise by 90°+Δθ2 degrees as shown in FIG.3.

During a subsequent stage S4, the first test flat is rotatedcounterclockwise from its position in stage S3 by 45° and the positionof the second test flat is unchanged. An optical path difference betweenthe first test flat and the second test flat is measured. An angle valueof a current position of the first test flat with respect to itsoriginal position in stage S1 is measured, which is then subtracted by45° to obtain an angular rotation error Δθ3 of stage S4. M₄ can becalculated as follows:

M ₄ =A ^(45°+Δθ3) +B ^(x)

where A^(45°+Δθ3) represents profile information of the first test flatafter being rotated clockwise from its original position in stage S1 by45°+Δθ3 degrees. B^(x) represents the profile information of the secondtest flat after its x-axis being reversed about the y-axis. M₄represents, after interference occurs between the first test flat andthe second test flat an optical path difference between the first testflat, after being rotated by 45°+Δθ3 degrees, as shown in FIG. 3.

During a subsequent stage S5, the first test flat is rotatedcounterclockwise from its position in stage S4 to its original positionin stage S1. The second test flat mirror is removed from the firstholder 5 and a third test flat mirror is placed on the first holder 5,so that the first test flat opposes the third test flat. An optical pathdifference between the first test flat and the third test flat ismeasured. An angular rotation error Δθ4 of a current position of thefirst test flat in stage S5 with respect to its original position instage S1 is measured. Then, M₅ can be calculated as follows:

M ₅ =A ^(Δθ4) +C ^(x),

where A^(Δθ4) represents profile information of the first test flatafter being rotated from its position in stage S4 to its originalposition in stage S1 with the angular rotation error Δθ4, C^(x)represents profile information of the third test flat after its x-axisbeing reversed about the y-axis; and M₅ represents, after interferenceoccurs between the first test flat and the third test flat, an opticalpath difference between the first test flat, after being rotated to itsoriginal position in stage S1 with the angular rotation error Δθ4, asshown in FIG. 3.

During stage S6, he first test flat mirror is removed from the secondholder 8, and the second test flat mirror is placed on the second holder8, so that the second test flat opposes the third test flat. An opticalpath difference M₆ between the second test flat and the third test flatis measured as:

M ₆ =B+C ^(x),

where B represents the profile information of the second test flat,C^(x) represents the profile information of the third test flat afterits x-axis being reversed about the y-axis; and M₆ represents an opticalpath difference between the second test flat and the third test flat, asshown in FIG. 3.

During stage S7, the third test flat mirror is removed from the firstholder 5, and the first test flat mirror is placed on the first holder5. A planar imaging unit 7 is inserted behind the first holder 5, sothat the planar imaging unit 7 is placed between the first test flat onthe first holder 5 and the second test flat on the second holder 8. Thesurfaces of the planar imaging unit 7 are generally parallel to thefirst test flat and the second test flat. An optical path differencebetween the first test flat and the second test flat is measured. FIG. 7schematically shows the interference measurement when the first testflat and the second test flat have a consistent x-axis direction.

An angular rotation error Δθ5 a of a current position of the first testflat in stage S7 with respect to its original position in stage S1 ismeasured. An angular rotation error Δθ5 b of a current position of thesecond test flat in stage S7 with respect to its original position instage S1 is measured.

M _(mid) =A ^(Δθ5a) +B ^(Δθ5b)

M_(mid) is assumed to be rotated an angular −Δθ5 b then

M ₇ =A ^(Δθ5) +B=M _(mid) ^(−Δθ5b) =A ^(Δθ5a−Δθ5b) +B^(Δθ5b−Δθ5b),Δθ5=Δθa−Δθb

M₇ can thus be calculated as:

M ₇ =A ^(Δθ5) +B,

where A^(Δθ5) represents profile information of the first test flat withits current position having an angular rotation error Δθ5 with respectto its original position in stage S1, B represents the profileinformation of the second test flat; and M₇ represents an optical pathdifference between the first test flat and the second test flat.

During another stage S8, based on the recorded optical path differencesM₁, M₂, M₃, M₄, M₅, M₆ and M₇ and the recorded rotation errors Δθ1, Δθ2,Δθ3, Δθ4 and Δθ5 the profile information of the first test flat, theprofile information of the second test flat, and the profile informationof the third test flat can be calculated by, for example, the computer12.

In the process described above, one planar imaging unit is added instage S7 to produce the measurement result M₇=A^(Δθ5)+B of A+B, and theangular error Δθ5 is measured.

Without being bound by theory, it is believed that in Cartesiancoordinates, a continuous function F(x, y) can be represented as a sumof an odd-odd function, an even-even function, an odd-even function, andan even-odd function:

F(x,y)=F _(ee) +F _(oo) +F _(oe) +F _(eo),

where x, y represent x, y coordinates of Cartesian coordinates, eerepresents an even-even component, oo represents an odd-odd component,eo represents an even-odd component, and oe represents an odd-evencomponent.

According to the characteristic of the odd-even function,

${{F_{ee}\left( {x,y} \right)} = \frac{{F\left( {x,y} \right)} + {F\left( {{- x},y} \right)} + {F\left( {x,{- y}} \right)} + {F\left( {{- x},{- y}} \right)}}{4}},{{F_{oo}\left( {x,y} \right)} = \frac{{F\left( {x,y} \right)} - {F\left( {{- x},y} \right)} - {F\left( {x,{- y}} \right)} + {F\left( {{- x},{- y}} \right)}}{4}},{{F_{oe}\left( {x,y} \right)} = \frac{{F\left( {x,y} \right)} - {F\left( {{- x},y} \right)} + {F\left( {x,{- y}} \right)} - {F\left( {{- x},{- y}} \right)}}{4}},{{F_{eo}\left( {x,y} \right)} = {\frac{{F\left( {x,y} \right)} + {F\left( {{- x},y} \right)} - {F\left( {x,{- y}} \right)} - {F\left( {{- x},{- y}} \right)}}{4}.}}$

Therefore, the planar profile information of the first test flat, thesecond test flat, and the third test flat can be deemed as a binaryfunction of the coordinates x, y. Accordingly, the planar profile of thefirst test flat, the second test flat, and the third test flat can berepresented by:

A=A _(ee) +A _(oe) +A _(eo) +A _(oo),

B=B _(ee) +B _(oe) +B _(eo) +B _(oo),

C=C _(ee) +C _(oe) +C _(eo) +C _(oo).

During detection, interference occurs between two flats opposing eachother, so one flat can be considered to be reversed. Assuming that theinterference occurs between two flats F(x, y) and G(x, y), two operators[ ]^(x) and [ ]^(θ) may be defined, where [ ]^(x) is a reverse operator,and [ ]^(θ) is a rotation operator:

[F(x,y)]^(x) =F(−x,y),

[F(x,y)]^(θ) =F(x cos θ−y sin θ,x sin θ+y sin θ).

Therefore,

[F(x,y)]^(180°) =F(−x,−y),

and thus

F(x,y)+G(−x,y)=F(x,y)+[G(x,y)]^(x).

By applying the two operators to the equations:

[F(x,y)]^(180°) =F _(ee) +F _(oo) −F _(oe) −F _(eo),

and

[F(x,y)]^(x) =F _(ee) −F _(oo) −F _(oe) +F _(eo).

The equations above shows that the even-odd function, the odd-evenfunction,

the even-even function, and the odd-odd function may be derived fromsimultaneous equations by means of rotation. It has been found that thefirst three functions can be easily derived by means of rotation, butthe odd-odd function is difficult to obtain.

In polar coordinates, the period of θ is 360°, and a periodic functioncan be represented by Fourier series. The period of the odd-odd functionin the polar coordinates is 180°. The function F_(oo) (x, y) may berepresented by a sum of Fourier series:

${{F_{oo}\left( {x,y} \right)} = {\sum\limits_{m = 1}{f_{2m}{\sin \left( {2m\; \theta} \right)}}}},$

where x²+y²=constant, f_(2m) is a corresponding coefficient, and thesubscript m is a natural number. In order to emphasize that the basefrequency of F_(oo)(x, y) is 2 (the period being 180°), the subscript ofF_(oo)(x,y) is replaced by 2θ, and the equations may be written as:

${F_{oo} = {F_{{oo},{2\theta}} = {F_{{oo},{2{odd}\; \theta}} + F_{{oo},{2{even}\; \theta}}}}},{F_{{oo},{2{even}\; \theta}} = {{\sum\limits_{m = {even}}{f_{2m}{\sin \left( {2m\; \theta} \right)}}} = {{\sum\limits_{m = 1}{f_{4m}{\sin \left( {4m\; \theta} \right)}}} = F_{{oo},{4\; \theta}}}}},{F_{{oo},{2{odd}\; \theta}} = {\sum\limits_{m = {odd}}{f_{2m}{{\sin \left( {2\; m\; \theta} \right)}.}}}}$

The subscript 4θ represents that the base frequency of is 4 (the periodbeing) 90°, as shown in the above equations, F_(oo,4θ) may be writtenas:

${F_{{oo},{4\theta}} = {F_{{oo},{4{odd}\; \theta}} + F_{{oo},{4{even}\; \theta}}}},{F_{{oo},{4{odd}\; \theta}} = {\sum\limits_{m = {odd}}{f_{4m}{\sin \left( {4\; m\; \theta} \right)}}}},{F_{{oo},{4{even}\; \theta}} = {\sum\limits_{m = {even}}{f_{4m}{{\sin \left( {4m\; \theta} \right)}.}}}}$

Therefore, an odd-odd function may be represented as a sum of a seriesof noddθ, where n=2, 4, 8, 16, 32, . . . .

F _(oo,2θ) =F _(oo,2oddθ) +F _(oo,4oddθ) +F _(oo,8oddθ) +F _(oo,16oddθ)+

By applying the rotation operator, the odd-odd function may berepresented by:

[F _(oo,2θ)]^(90°) =−F _(oo,2oddθ) +F _(oo,2evenθ),

[F _(oo,4θ)]^(45°) =−F _(oo,4oddθ) +F _(oo,4evenθ).

Therefore, the profile information may be written in the below form, inwhich the odd-odd function portion is replaced by its two frequencycomponents: 2oddθ and 4oddθ:

A=A _(ee) +A _(oe) +A _(eo) +A _(oo,2oddθ) +A _(oo,4oddθ),

B=B _(ee) +B _(oe) +B _(eo) +B _(oo,2oddθ) +B _(oo,4oddθ),

C=C _(ee) +C _(oe) +C _(eo) +C _(oo,2oddθ) +C _(oo,4oddθ).

In the equations, A_(ee) is the even-even component of the first testflat, A_(oe), is the odd-even component of the first test flat, A_(eo)is the even-odd component of the first test flat, A_(oo,2oddθ) is thepart with a base frequency of 2 in the odd-odd component of the firsttest flat, and A_(oo,4oddθ) is the part with a base frequency of 4 inthe odd-odd component of the first test flat; B_(ee) is the even-evencomponent of the second test flat, B_(oe) is the odd-even component ofthe second test flat, B_(eo) is the even-odd component of the secondtest flat, B_(oo,2oddθ) is the part with a base frequency of 2 in theodd-odd component of the second test flat, and B_(oo,4oddθ) is the partwith a base frequency of 4 in the odd-odd component of the second testflat; C_(ee) is the even-even component of the third test flat, C_(oe)is the odd-even component of the third test flat, C_(eo) is the even-oddcomponent of the third test flat, C_(oo,2oddθ) is the part with a basefrequency of 2 in the odd-odd component of the third test flat, andC_(oo,4oddθ) is the part with a base frequency of 4 in the odd-oddcomponent of the third test flat.

The foregoing measurement process may be represented by:

M ₁ =A+B ^(x),

M ₂ =A ^(180°+Δθ1) +B ^(x),

M ₃ =A ^(90°+Δθ2) +B ^(x),

M ₄ =A ^(45°+Δθ3) +B ^(x),

M ₅ =A ^(Δθ4) +C ^(x),

M ₆ =B+C ^(x),

M ₇ =A ^(Δθ5) +B,

where Δθ1, Δθ2, Δθ3, Δθ4 and Δθ5 are angular rotation errors, which aremeasured by the angular measurement instrument and stored.

The rotation absolute measurement process according to the presenttechnology is based on the angular measurement errors as below:

$\mspace{20mu} {{{A \approx \frac{A + A^{{\Delta \; \theta \; 5} - {\Delta \; \theta \; 4}}}{2}} = \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}},\mspace{20mu} {B^{x} = {{M_{1} - A} = {M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},\mspace{20mu} {A^{{180{^\circ}} + {\Delta \; \theta \; 1}} = {{M_{2} - B^{x}} = {M_{2} - M_{1} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},\mspace{20mu} {A^{180{^\circ}} = {\left( {M_{2} - M_{1}} \right)^{{- \Delta}\; \theta \; 1} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- {\Delta\theta}}\; 4} - {\Delta \; \theta \; 1}}}{2}}},\mspace{20mu} {M_{1} = {A + B_{x}}},{M_{2}^{\prime} = {{A^{180{^\circ}} + B^{x}} = {\left( {M_{2} - M_{1}} \right)^{{- \Delta}\; {\theta 1}} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 4} - {\Delta \; \theta \; 1}}}{2} + M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},{M_{3}^{\prime} = {{A^{90{^\circ}} + B^{x}} = {\left( {M_{3} - M_{1}} \right)^{{- \Delta}\; {\theta 2}} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 4} - {\Delta \; \theta \; 2}}}{2} + M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},{M_{4}^{\prime} = {{A^{45{^\circ}} + B^{x}} = {\left( {M_{4} - M_{1}} \right)^{{- \Delta}\; {\theta 3}} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 4} - {\Delta \; \theta \; 3}}}{2} + M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},{M_{5}^{\prime} = {\frac{M_{5} - M_{6} + M_{7}}{2} + \frac{M_{5} - {3\left( {M_{7} - M_{6}} \right)}}{4} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{\Delta\theta}\; 5} - {\Delta \; \theta \; 4}}}{4}}},\mspace{20mu} {M_{6} = {B + C^{x}}},}$

where M₁, M₂, M₃, M₄, M₅, M₆ and M₇ are optical path information; M₂′,M₃′, M₄′, and M₅′ are results obtained after angular error modification;and B^(x) represents the profile information after the x-axis of thetest flat is reversed about the y-axis.

The profile information A of the first test flat, the profileinformation B of the second test flat, and the profile information C ofthe third test flat can be calculated based on the results obtainedafter angular error modification as below:

$\mspace{20mu} {{{A_{oe} + A_{eo}} = {\left( {M_{1} - M_{2}^{\prime}} \right)/2}},\mspace{20mu} {{B_{oe} + B_{eo}} = \left\{ {{\left\lbrack {M_{1} - \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} - \left( {A_{oe} + A_{eo}} \right)} \right\}^{x}},\mspace{20mu} {{C_{oe} + C_{eo}} = \left\{ {{\left\lbrack {M_{5}^{\prime} - \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} - \left( {A_{oe} + A_{eo}} \right)} \right\}^{x}},{A_{ee} = {\begin{Bmatrix}{{\left\lbrack {M_{1} + \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} - {\left\lbrack {M_{6} + \left( M_{6} \right)^{180{^\circ}}} \right\rbrack/2} +} \\\left( {{\left\lbrack {M_{1} + \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} - {\left\lbrack {M_{6} + \left( M_{6} \right)^{180{^\circ}}} \right\rbrack/2}} \right)^{x}\end{Bmatrix}/4}},\mspace{20mu} {B_{ee} = {\left\{ {{\left\lbrack {M_{1} + \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{1} + \left( M_{1} \right)^{180{^\circ}}} \right\rbrack^{x}/2} - {2A_{ee}}} \right\}/2}},\mspace{20mu} {C_{ee} = {\left\{ {{\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack^{x}/2} - {2A_{ee}}} \right\}/2}},{A_{{oo},{2{odd}\; \theta}} = {\left\lbrack {M_{1} - \left( {A_{oe} + A_{eo} + A_{ee}} \right) - M_{3}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee}} \right)^{90{^\circ}}} \right\rbrack/2}},{B_{{oo},{2{odd}\; \theta}} = {\begin{Bmatrix}{\left\lbrack {M_{1} - \left( {A_{oe} + A_{eo} + A_{ee}} \right) - \left( {B_{oe} + B_{eo} + B_{ee}} \right)^{x}} \right\rbrack^{90{^\circ}} -} \\{M_{3}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee}} \right) + \left( {B_{oe} + B_{eo} + B_{ee}} \right)^{x}}\end{Bmatrix}/2}},{C_{{oo},{2{odd}\; \theta}} = {\begin{Bmatrix}{\left\lbrack {M_{6} - \left( {B_{oe} + B_{eo} + B_{ee}} \right) - \left( {C_{oe} + C_{eo} + C_{ee}} \right)} \right\rbrack^{90{^\circ}} -} \\{M_{6} + \left( {B_{oe} + B_{eo} + B_{ee}} \right) + \left( {C_{oe} + C_{eo} + C_{ee}} \right) + {2B_{{oo},{2\; {odd}\; \theta}}}}\end{Bmatrix}/2}},{A_{{oo},{4{odd}\; \theta}} = {\left\lbrack {M_{1} - \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2{odd}\; \theta}}} \right) - M_{4}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2{odd}\; \theta}}} \right)^{45{^\circ}}} \right\rbrack/2}}}$$\mspace{20mu} {B_{{oo},{4{odd}\; \theta}} = {\begin{Bmatrix}{\begin{bmatrix}{M_{1} - \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2\; {odd}\; \theta}}} \right) -} \\\left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right)^{x}\end{bmatrix}^{45{^\circ}} -} \\\begin{matrix}{M_{4}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2\; {odd}\; \theta}}} \right)^{45{^\circ}} +} \\\left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right)^{x}\end{matrix}\end{Bmatrix}/2}}$$\mspace{20mu} {C_{{oo},{4{odd}\; \theta}} = {\left\{ \begin{matrix}{\begin{bmatrix}{M_{6} - \left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right) -} \\\left( {C_{oe} + C_{eo} + C_{ee} + C_{{oo},{2\; {odd}\; \theta}}} \right)^{x}\end{bmatrix}^{45{^\circ}} -} \\\begin{matrix}{M_{6} + \left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right) +} \\\left( {C_{oe} + C_{eo} + C_{ee} + C_{{oo},{2{odd}\; \theta}}} \right)^{x}\end{matrix}\end{matrix} \right\} / 2}}$

Based on the above results, we may obtain:

A=A _(ee) +A _(oe) +A _(eo) +A _(oo,2oddθ) +A _(oo,4oddθ),

B=B _(ee) +B _(oe) +B _(eo) +B _(oo,2oddθ) +B _(oo,4oddθ),

C=C ₃₃ +C _(oe) +C _(eo) +C _(oo,2oddθ) +C _(oo,4oddθ).

In the equations, A_(ee) is the even-even component of the first testflat, A_(oe) is the odd-even component of the first test flat, A_(eo) isthe even-odd component of the first test flat, A_(oo,2oddθ) is the partwith a base frequency of 2 in the odd-odd component of the first testflat, and A_(oo,4oddθ) is the part with a base frequency of 4 in theodd-odd component of the first test flat. B_(ee) is the even-evencomponent of the second test flat, B_(oe) is the odd-even component ofthe second test flat, B_(eo) is the even-odd component of the secondtest flat, B_(oo,2oddθ) is the part with a base frequency of 2 in theodd-odd component of the second test flat, and B_(oo,4oddθ) is the partwith a base frequency of 4 in the odd-odd component of the second testflat. C_(ee) is the even-even component of the third test flat, C_(oe)is the odd-even component of the third test flat, C_(eo) is the even-oddcomponent of the third test flat, C_(oo,2oddθ) is the part with a basefrequency of 2 in the odd-odd component of the third test flat, andC_(oo,4oddθ) is the part with a base frequency of 4 in the odd-oddcomponent of the third test flat.

M₁, M₂, M₃, M₄, M₅, M₆ and M₇ are optical path information; M₂′, M₃′,M₄′, and M₅′ are results obtained after angular error modification; andB^(x) represents the profile information after the x-axis of the testflat is reversed about the y-axis. For the function F(x,y), twooperators [ ]^(x) and [ ]^(θ) are defined, wherein [ ]^(x) is a reverseoperator, and [ ]^(θ) is a rotation operator:

[F(x,y)]^(x) =F(−x,y),

[F(x,y)]^(θ) =F(x cos θ−y sin θ,x sin θ+y cos θ).

Therefore,

[F(x,y)]^(180°) =F(−x,−y),

and thus

F(x,y)+G(−x,y)=F(x,y)+[G(x,y)]^(x).

By applying the two operators to the equations we may obtain:

[F(x,y)]^(180°) =F _(ee) +F _(oo) −F _(oe) −F _(eo),

and

[F(x,y)]^(x) =F _(ee) −F _(oo) −F _(oe) +F _(eo).

The definition to the operation of the function F may be applied to thefirst test flat, the second test flat, the third test flat, acombination thereof, a combination of frequency components thereof, acombination of optical path difference information M₁, M₂, M₃, M₄, M₅,M₆ and M₇, and/or a combination of optical path difference informationM₂′, M₃′, M₄′, and M₅′ obtained after angular error modification.

From the process described above, it can be seen that the error causedby the rotation angle changes to Δθ5−Δθ4 from Δθ1, Δθ2, Δθ3, Δθ4. Themeasurement accuracy can thus be improved.

The process results are simulated as shown in FIGS. 10 a, 10 b, and 10c. The first test flat shown in FIG. 10 a, the second test flat shown inFIG. 10 b, and the third test flat shown in FIG. 10 c are generated by36 Zernike coefficients. A program listing for performing a Zernike fitis shown in the Appendix. Assuming the angular rotation errors areΔθ1=0.8°, Δθ2=0.1°, Δθ3=0.5°, Δθ4=0.7°, Δθ5=0.9°, the results of theforegoing measurement process are shown as detection errors of the firsttest flat compared with the original profile shown in FIG. 11 a-11 c.The results of the prior art six-stage measurement process are shown inFIG. 12 a-12 c.

A program listing for implementing a Zernike fit flat process, certainembodiments of the process described above, and the rotation process inMatlab are included in the Appendices I-III, which forms part of thepresent disclosure. In other embodiments, the foregoing processes may beimplemented in other suitable languages and/or systems.

From the foregoing, it will be appreciated that specific embodiments ofthe disclosure have been described herein for purposes of illustration,but that various modifications may be made without deviating from thedisclosure. In addition, many of the elements of one embodiment may becombined with other embodiments in addition to or in lieu of theelements of the other embodiments. Accordingly, the technology is notlimited except as by the appended claims.

APPENDIX I Zernike Fit process function shap=zernike_36(a) wl=632.8e−3;P_V=0.01*wl; znikexs=a; D=50; w_r=361; w_a =3600; for m=1:1:w_r  forn=1:1:w_a % r=m/(w_r−1)−1/(w_r−1); a=n*(pi*2/(w_a)); z(1)=1;z(2)=r*cos(a); z(3)=r*sin(a); z(4)=2*r{circumflex over ( )}2−1;z(5)=r{circumflex over ( )}2*cos(2*a); z(6)=r{circumflex over( )}2*sin(2*a); z(7)=(3*r{circumflex over ( )}2−2)*r*cos(a);z(8)=(3*r{circumflex over ( )}2−2)*r*sin(a); z(9)=6*r{circumflex over( )}4−6*r{circumflex over ( )}2+1; z(10)=r{circumflex over( )}3*cos(3*a); z(11)=r{circumflex over ( )}3*sin(3*a);z(12)=(4*r{circumflex over ( )}2−3)*r{circumflex over ( )}2*cos(2*a);z(13)=(4*r{circumflex over ( )}2−3)*r{circumflex over ( )}2*sin(2*a);z(14)=(10*r{circumflex over ( )}4−12*r{circumflex over( )}2+3)*r*cos(a); z(15)=(10*r{circumflex over ( )}4−12*r{circumflexover ( )}2+3)*r*sin(a); z(16)=20*r{circumflex over ( )}6−30*r{circumflexover ( )}4+12*r{circumflex over ( )}2−1; z(17)=r{circumflex over( )}4*cos(4*a); z(18)=r{circumflex over ( )}4*sin(4*a);z(19)=(5*r{circumflex over ( )}2−4)*r{circumflex over ( )}3*cos(3*a);z(20)=(5*r{circumflex over ( )}2−4)*r{circumflex over ( )}3*sin(3*a);z(21)=(15*r{circumflex over ( )}4−20*r{circumflex over( )}2+6)*r{circumflex over ( )}2*cos(2*a); z(22)=(15*r{circumflex over( )}4−20*r{circumflex over ( )}2+6)*r{circumflex over ( )}2*sin(2*a);z(23)=(35*r{circumflex over ( )}6−60*r{circumflex over( )}4+30*r{circumflex over ( )}2−4)*r*cos(a); z(24)=(35*r{circumflexover ( )}6−60*r{circumflex over ( )}4+30*r{circumflex over( )}2−4)*r*sin(a); z(25)=70*r{circumflex over ( )}8−140*r{circumflexover ( )}6+90*r{circumflex over ( )}4−20*r{circumflex over ( )}2+1;z(26)=r{circumflex over ( )}5*cos(5*a); z(27)=r{circumflex over( )}5*sin(5*a); z(28)=(6*r{circumflex over ( )}2−5)*r{circumflex over( )}4*cos(4*a); z(29)=(6*r{circumflex over ( )}2−5)*r{circumflex over( )}4*sin(4*a); z(30)=(21*r{circumflex over ( )}4−30*r{circumflex over( )}2+10)*r{circumflex over ( )}3*cos(3*a); z(31)=(21*r{circumflex over( )}4−30*r{circumflex over ( )}2+10)*r{circumflex over ( )}3*sin(3*a);z(32)=(56*r{circumflex over ( )}6−105*r{circumflex over( )}4+60*r{circumflex over ( )}2−10)*r{circumflex over ( )}2*cos(2*a);z(33)=(56*r{circumflex over ( )}6−105*r{circumflex over( )}4+60*r{circumflex over ( )}2−10)*r{circumflex over ( )}2*sin(2*a);z(34)=(126*r{circumflex over ( )}8−280*r{circumflex over( )}6+210*r{circumflex over ( )}4−60*r{circumflex over( )}2+5)*r*cos(a); z(35)=(126*r{circumflex over ( )}8−280*r{circumflexover ( )}6+210*r{circumflex over ( )}4−60*r{circumflex over( )}2+5)*r*sin(a); z(36)=252*r{circumflex over ( )}10−630*r{circumflexover ( )}8+560*r{circumflex over ( )}6−210*r{circumflex over( )}4+30*r{circumflex over ( )}2−1; sharp(m,n)=z*znikexs';  end endp_v=max(max(sharp))−min(min(sharp)); I=cos(sharp*2*pi/wl); shap=sharp;

APPENDIX II Main process clear clca=[0,0,0,0,0,0,0,0,0,0.0005,0,0,0,0,0,0,0,0.0005,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];surfa=zernike_36(a);b=[0,0,0,0,0,0,0,0,0,0,0.0005,0,0,0,0,0,0,0.0005,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];surfb=zernike_36(b);c=[0,0,0,0,0,0,0,0,0,0,0,0,0.0005,0,0,0,0,0,0.0005,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];surfc=zernike_36(c); dirt1=8; dirt2=1; dirt3=5; dirt4=7; dirt5=9;rotation180=1800; rotation90=900; rotation45=450;surfa180=rotation(surfa,(rotation180+dirt1));surfa90=rotation(surfa,(rotation90+dirt2));surfa45=rotation(surfa,(rotation45+dirt3)); surfbback=back(surfb);surfcback=back(surfc); M11=surfa+surfbback; M21=surfa180+surfbback;M31=surfa90+surfbback; M41=surfa45+surfbback;M51=rotation(surfa,dirt4)+surfcback; M61=surfb+surfcback;M71=rotation(surfa,dirt5)+surfb; A=rotation((M51−M61+M71)/2,3600−dirt4);Bx=M11−A; A180dirt1=M21−Bx; A180=rotation(A180dirt1,3600−dirt1); M1=M11;M2=A180+Bx; M3=rotation((M31−Bx),3600−dirt2)+Bx;M4=rotation((M41−Bx),3600−dirt3)+Bx;M5=(M51−M61+M71)/2+(M51−3*(M71−M61))/4+rotation((M51−M61+M71)/4,(dirt5−dirt4)); M6=M61; aoeaeo=(M1−M2)/2;boebeo=back(((M1−rotation(M1,rotation180))/2−aoeaeo));coeceo=back(((M5−rotation(M5,rotation180))/2−aoeaeo));m1=(M1+rotation(M1,rotation180))/2; m5=(M5+rotation(M5,rotation180))/2;m6=(M6+rotation(M6,rotation180))/2; aee=(m1+m5−m6+back((m1+m5−m6)))/4;bee=(m1+back(m1)−2*aee)/2; cee=(m5+back(m5)−2*aee)/2;m11=M1−(aoeaeo+aee)−back((boebeo+bee));m13=M3−rotation((aoeaeo+aee),rotation90)−back((boebeo+bee));m16=M6−(boebeo+bee)−back((coeceo+cee)); aoo2odd=(m11−m13)/2;boo2odd=(rotation(m11,rotation90)−m13)/2;coo2odd=(rotation(m16,rotation90)−m16+rotation(m11,rotation90)−m13)/2;m11p=M1−(aoeaeo+aee+aoo2odd)−back((boebeo+bee+boo2odd));m41p=M4−rotation((aoeaeo+aee+aoo2odd),rotation45)−back((boebeo+bee+boo2odd));m61p=M6−(boebeo+bee+boo2odd)−back((coeceo+cee+coo2odd));aoo4odd=(m11p−m41p)/2; boo4odd=(rotation(m11p,rotation45)−m41p)/2;coo4odd=(rotation(m61p,rotation45)−m61p+rotation(m11p,rotation45)−m41p)/2;finala=aee+aoeaeo+aoo2odd+aoo4odd; finalb=bee+boebeo+boo2odd+boo4odd;finalc=cee+coeceo+coo2odd+coo4odd;

APPENDIX III Rotation process function shap=rotation(sharp,th)  forri=1:1:361   for aj=1:1:3600    if (aj+th)<=3600    zrotation(ri,aj+th)=sharp(ri,aj);    else    zrotation(ri,aj+th−3600)=sharp(ri,aj);    end   end endshap=zrotation;

I/We claim:
 1. An apparatus for detecting optical profile, comprising: alaser for emitting a laser beam; a beam splitter for transmitting thelaser beam emitted by the laser as a illuminating light and a reflectinginterfered test light; a collimation optical unit for forming a uniformilluminated area with the laser beam emitted by the laser; first andsecond holders respectively holding a first test flat mirror and asecond test flat mirror; a phase shifter connected with the firstholder; an angular measurement unit for measuring an angular error ofthe first test flat mirror and the second test flat mirror on the twoholders, wherein the first test flat mirror has a first test flat andthe second test flat mirror has a second test flat; a planar imagingunit for generating the interfered test light having a directiongenerally along an x-axis direction of the first test flat and an x-axisdirection of the second test flat; a convergence optical unit forprojecting the interfered test light onto a detector, wherein: the laseris placed on a front focus of the beam splitter and the collimationoptical unit; the first holder is placed proximate the collimationoptical unit, and has its center of optical axis generally aligned withthe center of the collimation optical unit and the center of the beamsplitter; the second holder is placed proximate and generally parallelto the first holder; the planar imaging unit is placed between the firsttest flat on the first holder and the second test flat on the secondholder, a face of the planar imaging unit is generally parallel to thefirst test flat and the second test flat; and the beam splitter, theconvergence optical unit, and the detector is arranged in series.
 2. Theapparatus of claim 1, wherein the laser is a monochromatic light sourceof visible light, ultraviolet light, deep ultraviolet light, or extremeultraviolet light.
 3. The apparatus of claim 1, wherein the beamsplitter includes a prism or a polarizer.
 4. The apparatus of claim 1,wherein the planar imaging unit includes a flat mirror or a totalreflection prism.
 5. The apparatus of claim 1, wherein the first holderis a 5-dimensional optical adjustable holder, a 6-dimensional opticaladjustable holder, or a 8-dimensional optical adjustable holder capableof angular rotation.
 6. The apparatus of claim 1, wherein the angularmeasurement unit includes a goniometer.
 7. A method for detecting anoptical profile, comprising: providing a first test flat mirror having afirst test flat, a second test flat mirror having a second test flat,and a third test flat mirror having a third test flat; placing the firsttest flat mirror in the second holder and placing the second test flatmirror in the first holder so that the first test flat faces the secondtest flat; setting the first test flat and the second test flat tooppose each other and recording position information of these two testflats in an x-axis direction and a y-axis direction; defining a currentposition of the first test flat as its original position; measuring anoptical path difference between the first test flat and the second testflat, calculating an optical path difference as:M ₁ =A+B ^(x), where A represents profile information of the first testflat, B represents profile information of the second test flat; thefirst test flat and the second test flat are functions of x, ycoordinates, A=A(x,y), B=B(x,y); M₁ represents an optical pathdifference between the first test flat and the second test flat in thefirst interference measurement; rotating the first test flat mirrorclockwise from its original position on the second holder in stage S1 by180° and maintaining the position of the second test flat unchanged;measuring an optical path difference between the first test flat and thesecond test flat, measuring an angle value of a current position of thefirst test flat with respect to its position to obtain an angularrotation error Δθ1; calculating M₂ asM ₂ =A ^(180°+Δθ1) +B ^(x), where A^(180°+Δθ1) represents profileinformation of the first test flat after being rotated clockwise by180°+Δθ1 degrees, B^(x) represents the profile information of the secondtest flat after its x-axis being reversed about the y-axis; and M₂represents, after interference occurs between the first test flat andthe second test flat, an optical path difference between the first testflat, after being rotated clockwise by 180°+Δθ1 degrees from itsposition; rotating the first test flat counterclockwise from itsposition in stage S2 by 90° and maintaining the position of the secondtest flat unchanged, namely, rotating the first test flat clockwise fromits original position by 90°; measuring an optical path differencebetween the first test flat and the second test flat, whereincalculating an angle value of a current position of the first test flatwith respect to its original position by subtracting 90° to obtain anangular rotation error Δθ2; calculating M₃ asM ₃ =A ^(90°+Δθ2) +B ^(x), where A^(90°+Δθ2)+B^(x) represents profileinformation of the first test flat after being rotated clockwise fromits original position by 90°+Δθ2 degrees, B^(x) represents the profileinformation of the second test flat after its x-axis being reversedabout the y-axis; and M₃ represents, after interference occurs betweenthe first test flat and the second test flat, an optical path differencebetween the first test flat, after being rotated clockwise by 90°+Δθ2degrees, and the second test flat; rotating the first test flatcounterclockwise from its position by 45° and maintaining the positionof the second test flat unchanged; measuring an optical path differencebetween the first test flat and the second test flat, calculating anangle value of a current position of the first test flat with respect toits original position by subtracting 45° to obtain an angular rotationerror Δθ3; calculating M₄ asM ₄ =A ^(45°+Δθ3) +B ^(x), where A^(45°+Δθ3) represents profileinformation of the first test flat after being rotated clockwise fromits original position by 45°+Δθ3 degrees, B^(x) represents the profileinformation of the second test flat after its x-axis being reversedabout the y-axis; and M₄ represents, after interference occurs betweenthe first test flat and the second test flat, an optical path differencebetween the first test flat, after being rotated clockwise by 45°+Δθ3degrees; rotating the first test flat counterclockwise from its positionto its original position; removing the second test flat mirror from thefirst holder and placing a third test flat mirror on the first holder,so that the first test flat opposes the third test flat; measuring anoptical path difference between the first test flat and the third testflat, calculating an angular rotation error Δθ94 of a current positionof the first test flat with respect to its original position;calculating M₅ asM ₅ =A ^(Δθ4) +C ^(x), where A^(Δθ4) represents profile information ofthe first test flat after being rotated from its position in stage S4 toits original position in stage S1 with the angular rotation error Δθ4,C^(x) represents profile information of the third test flat after itsx-axis being reversed about the y-axis; and M₅ represents, afterinterference occurs between the first test flat and the third test flat,an optical path difference between the first test flat, after beingrotated to its original position with the angular rotation error Δθ4;removing the first test flat mirror from the second holder, and placingthe second test flat mirror on the second holder, so that the secondtest flat opposes the third test flat; measuring an optical pathdifference M₆ between the second test flat and the third test flat asM ₆ =B+C ^(x), where B represents the profile information of the secondtest flat, C^(x) represents the profile information of the third testflat after its x-axis being reversed about the y-axis; and M₆represents, after interference occurs between the second test flat andthe third test flat, an optical path difference between the second testflat and the third test flat; removing the third test flat mirror fromthe first holder, and placing the first test flat mirror on the firstholder; placing a planar imaging unit between the first holder and thesecond holder, so that adjacent surfaces of the first test flat on thefirst holder and the planar imaging unit are kept to be parallel, andadjacent surfaces of the planar imaging unit and the second test flat onthe second holder are kept to be parallel; measuring an optical pathdifference between the first test flat and the second test flat,calculating an angular rotation error Δθ5 of current positions of thefirst test flat with respect to its original position; calculating M₇ asM ₇ =A ^(Δθ5) +B, where A^(Δθ5) represents profile information of thefirst test flat with its current position having an angular rotationerror Δθ5 with respect to its original position, B represents theprofile information of the second test flat; and M₇ represents anoptical path difference between the first test flat with its currentposition having an angular rotation error Δθ5 with respect to itsoriginal position and the second test flat; based on the recordedoptical path differences M₁, M₂, M₃, M₄, M₅, M₆ and M₇, and the recordedrotation errors Δθ1, Δθ2, Δθ3, Δθ4 and Δθ5, calculating the profileinformation A of the first test flat, the profile information B of thesecond test flat, and the profile information C of the third test flat.8. The method of claim 7, wherein the profile information A of the firsttest flat, the profile information B of the second test flat, and theprofile information C of the third test flat are represented by:A=A _(ee) +A _(oe) +A _(eo) +A _(oo,2oddθ) +A _(oo,4oddθ),B=B _(ee) +B _(oe) +B _(eo) +B _(oo,2oddθ) +B _(oo,4oddθ),C=C _(ee) +C _(oe) +C _(eo) +C _(oo,2oddθ) +C _(oo,4oddθ), where A_(ee)is an even-even component of the first test flat, A_(oe) is an odd-evencomponent of the first test flat, A_(eo) is an even-odd component of thefirst test flat, A_(oo,2oddθ) is a part with a base frequency of 2 in anodd-odd component of the first test flat, and A_(oo,4oddθ) is a partwith a base frequency of 4 in the odd-odd component of the first testflat; B_(ee) is an even-even component of the second test flat, B_(oe)is an odd-even component of the second test flat, B_(eo) is an even-oddcomponent of the second test flat, B_(oo,2oddθ) is a part with a basefrequency of 2 in an odd-odd component of the second test flat, andB_(oo,4oddθ) is a part with a base frequency of 4 in an odd-oddcomponent of the second test flat; C_(ee) is an even-even component ofthe third test flat, C_(oe) is an odd-even component of the third testflat, C_(eo) is an even-odd component of the third test flat,C_(oo,2oddθ) is a part with a base frequency of 2 in an odd-oddcomponent of the third test flat, and C_(oo,4oddθ) is a part with a basefrequency of 4 in an odd-odd component of the third test flat.
 9. Themethod of claim 7, further comprising based on the angular measurementerrors Δθ1, Δθ2, Δθ3, Δθ4 and Δθ5 calculating optical path informationas below:$\mspace{85mu} {{{A \approx \frac{A + A^{{\Delta \; \theta \; 5} - {\Delta \; \theta \; 4}}}{2}} = \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}},\mspace{79mu} {B^{x} = {{M_{1} - A} = {M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},\mspace{79mu} {A^{{180{^\circ}} + {{\Delta\theta}\; 1}} = {{M_{2} - B^{x}} = {M_{2} - M_{1} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},\mspace{79mu} {A^{180{^\circ}} = {\left( {M_{2} - M_{1}} \right)^{{- \Delta}\; \theta \; 1} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 4} - {\Delta \; \theta \; 1}}}{2}}},\mspace{79mu} {M_{1} = {A + B_{x}}},{M_{2}^{\prime} = {A^{180{^\circ}} + {B^{x}\left( {M_{2} - M_{1}} \right)}^{{- \Delta}\; \theta \; 1} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 4} - {\Delta \; \theta \; 1}}}{2} + M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}},{M_{3}^{\prime} = {{A^{90{^\circ}} + B^{x}} = {\left( {M_{3} - M_{1}} \right)^{{- \Delta}\; \theta \; 2} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 4} - {\Delta \; \theta \; 2}}}{2} + M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},{M_{4}^{\prime} = {{A^{45{^\circ}} + B^{x}} = {\left( {M_{4} - M_{1}} \right)^{{- \Delta}\; \theta \; 3} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 4} - {\Delta \; \theta \; 3}}}{2} + M_{1} - \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{- \Delta}\; \theta \; 4}}{2}}}},{M_{5}^{\prime} = {\frac{M_{5} - M_{6} + M_{7}}{2} + \frac{M_{5} - {3\left( {M_{7} + M_{6}} \right)}}{4} + \frac{\left( {M_{5} - M_{6} + M_{7}} \right)^{{{- \Delta}\; \theta \; 5} - {\Delta \; {\theta 4}}}}{4}}},\mspace{79mu} {M_{6} = {B + C^{x}}},}$where M₁, M₂, M₃, M₄, M₅, M₆ and M₇ are optical path information; M₂′,M₃′, M₄′, and M₅′ are results obtained after angular error modification;and B^(x) represents the profile information after the x-axis of thesecond test flat is reversed about the y-axis.
 10. The method of claim9, wherein: the profile information A of the first test flat, theprofile information B of the second test flat, and the profileinformation C of the third test flat are calculated based on the resultsM₂′, M₃′, M₄′, and M₅′ obtained after the angular error modification asbelow:$\mspace{20mu} {{{A_{oe} + A_{eo}} = {\left( {M_{1} - M_{2}^{\prime}} \right)/2}},\mspace{20mu} {{B_{oe} + B_{eo}} = \left\{ {{\left\lbrack {M_{1} - \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} - \left( {A_{oe} + A_{eo}} \right)} \right\}^{x}},\mspace{20mu} {{C_{oe} + C_{eo}} = \left\{ {{\left\lbrack {M_{5}^{\prime} - \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} - \left( {A_{oe} + A_{eo}} \right)} \right\}^{x}},{A_{ee} = {\begin{Bmatrix}{{\left\lbrack {M_{1} + \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} - {\left\lbrack {M_{6} + \left( M_{6} \right)^{180{^\circ}}} \right\rbrack/2} +} \\\left( {{\left\lbrack {M_{1} + \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} - {\left\lbrack {M_{6} + \left( M_{6} \right)^{180{^\circ}}} \right\rbrack/2}} \right)^{x}\end{Bmatrix}/4}},\mspace{20mu} {B_{ee} = {\left\{ {{\left\lbrack {M_{1} + \left( M_{1} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{1}\left( M_{1} \right)}^{180{^\circ}} \right\rbrack^{x}/2} - {2A_{ee}}} \right\}/2}},\mspace{20mu} {C_{ee} = {\left\{ {{\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack/2} + {\left\lbrack {M_{5}^{\prime} + \left( M_{5}^{\prime} \right)^{180{^\circ}}} \right\rbrack^{x}/2} - {2A_{ee}}} \right\}/2}},{A_{{oo},{2{odd}\; \theta}} = {\left\lbrack {M_{1} - \left( {A_{oe} + A_{eo} + A_{ee}} \right) - M_{3}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee}} \right)^{90{^\circ}}} \right\rbrack/2}},{B_{{oo},{2{odd}\; \theta}} = {\left\{ {\left\lbrack {M_{1} - \left( {A_{oe} + A_{eo} + A_{ee}} \right) - \left( {B_{oe} + B_{eo} + B_{ee}} \right)^{x}} \right\rbrack^{90{^\circ}} - M_{3}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee}} \right) + \left( {B_{oe} + B_{eo} + B_{ee}} \right)^{x}} \right\}/2}},{C_{{oo},{2{odd}\; \theta}} = {\left\{ {\left\lbrack {M_{6} - \left( {B_{oe} + B_{eo} + B_{ee}} \right) - \left( {C_{oe} + C_{eo} + C_{ee}} \right)} \right\rbrack^{90{^\circ}} - M_{6} + \left( {B_{oe} + B_{eo} + B_{ee}} \right) + \left( {C_{oe} + C_{eo} + C_{ee}} \right) + {2B_{{{oo}\; 2},{{odd}\; \theta}}}} \right\}/2}},{A_{{oo},{4{odd}\; \theta}} = {\left\lbrack {M_{1} - \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2{odd}\; \theta}}} \right) - M_{4}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2{odd}\; \theta}}} \right)^{45{^\circ}}} \right\rbrack/2}}}$$\mspace{20mu} {{B_{{oo},{4{odd}\; \theta}} = {\left\{ \begin{matrix}{\begin{bmatrix}{M_{1} - \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2{odd}\; \theta}}} \right) -} \\\left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right)^{x}\end{bmatrix}^{45{^\circ}} -} \\\begin{matrix}{M_{4}^{\prime} + \left( {A_{oe} + A_{eo} + A_{ee} + A_{{oo},{2{odd}\; \theta}}} \right)^{45{^\circ}} +} \\\left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right)^{x}\end{matrix}\end{matrix} \right\} / 2}},\mspace{20mu} {C_{{oo},{4{odd}\; \theta}} = {\begin{Bmatrix}{\begin{bmatrix}{M_{6} - \left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right) -} \\\left( {C_{oe} + C_{eo} + C_{ee} + C_{{oo},{2{odd}\; \theta}}} \right)^{x}\end{bmatrix}^{45{^\circ}} -} \\\begin{matrix}{M_{6} + \left( {B_{oe} + B_{eo} + B_{ee} + B_{{oo},{2{odd}\; \theta}}} \right) +} \\\left( {C_{oe} + C_{eo} + C_{ee} + C_{{oo},{2\; {odd}\; \theta}}} \right)^{x}\end{matrix}\end{Bmatrix}/2}},}$ based on the above results, calculating as follows:A=A _(ee) +A _(oe) +A _(eo) +A _(oo,2oddθ) +A _(oo,4oddθ),B=B _(ee) +B _(oe) +B _(eo) +B _(oo,2oddθ) +B _(oo,4oddθ),C=C _(ee) +C _(oe) +C _(eo) +C _(oo,2oddθ) +C _(oo,4oddθ), where A_(ee)is an even-even component of the first test flat, A_(oe) is an odd-evencomponent of the first test flat, A_(eo) is an even-odd component of thefirst test flat, A_(oo,2oddθ) is a part with a base frequency of 2 in anodd-odd component of the first test flat, and A_(oo,4oddθ) is a partwith a base frequency of 4 in the odd-odd component of the first testflat; B_(ee) is an even-even component of the second test flat, B_(oe)is an odd-even component of the second test flat, B_(eo) is an even-oddcomponent of the second test flat, B_(oo,2oddθ) is a part with a basefrequency of 2 in an odd-odd component of the second test flat, andB_(oo,4oddθ) is a part with a base frequency of 4 in an odd-oddcomponent of the second test flat; C_(ee) is an even-even component ofthe third test flat, C_(oe) is an odd-even component of the third testflat, C_(eo) is an even-odd component of the third test flat,C_(oo,2oddθ) is a part with a base frequency of 2 in an odd-oddcomponent of the third test flat, and C_(oo,4oddθ) is a part with a basefrequency of 4 in an odd-odd component of the third test flat; M₁, M₂,M₃, M₄, M₅, M₆ and M₇ are optical path information; M₂′, M₃′, M₄′, andM₅′ are results obtained after angular error modification; and B^(x)represents the profile information after the x-axis of the test flat isreversed about the y-axis.